Chapter 9 - Dynamical Decoupling Controls

Introduction

In the last chapter, it was shown that a symmetry in the system-bath Hamiltonian, if present, could be used to construct states immune to noise. Under certain conditions, it is possible to remove errors or to create a symmetry in the evolution of a quantum system. This is done by averaging away errors though repeated use of external controls which act on the system. These controls are often called "dynamical decoupling controls" due to their original objective of decoupling the system from the bath. They are quite generally useful controls to consider for the elimination and/or reduction of errors. In this chapter, a simple introduction to dynamical decoupling controls is given and some important concepts discussed.

General Setting

As stated in Chapter 8 the Hamiltonian describing the evolution of a system and bath which are coupled together can always be written as

$H=H_{S}\otimes I_{B}+I_{S}\otimes H_{B}+H_{I},\,\!$

where $H_{S}\,\!$ acts only on the system, $H_{B}\,\!$ acts only on the bath, and

$H_{I}=\sum _{\alpha }S_{\alpha }\otimes B_{\alpha },\,\!$

is the interaction Hamiltonian with the $S_{\alpha }\,\!$ acting only on the system and the $B_{\alpha }\,\!$ acting only on the bath.

The idea is to modify the evolution of the system and bath such that the errors are reduced or eliminated. A fairly general starting point to see how this is done is the so-called Magnus expansion. Let

The Single-Qubit Case

The simplest case involves the elimination of an error on a single qubit state.